Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields - Böckle, Gebhard; Juschka, Ann-Kristin (Forum of mathematics. Sigma, 2023)
Let K be a finite extension of the p-adic field Q𝑝 of degree d, let F be a finite field of characteristic p and let 𝐷 be an n-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of K over the field F. For the universal mod p pseudodeformation ring 𝑅univ 𝐷 of 𝐷, we prove the following: The ring 𝑅ps 𝐷 is equidimensional of dimension 𝑑𝑛2 + 1. Its reduced quotient 𝑅univ 𝐷,red contains a dense open subset of regular points x whose associated pseudocharacter 𝐷 𝑥 is absolutely irreducible and nonspecial in a certain technical sense that we shall define. Moreover, we will characterize in most cases when K does not contain a p-th root of unity the singular locus of Spec 𝑅univ 𝐷 . Similar results were proved by Chenevier for the generic fiber of the universal pseudodeformation ring 𝑅univ 𝐷 of 𝐷.